Algebra step by step

Linear & inequalities — brackets, fractions (cleared with the lowest common denominator) and letters on both sides, finishing with a check that puts the answer back in. Inequalities say when the sign flips (multiplying or dividing by a negative), and handle compound inequalities (-3 < 2x + 1 ⇐ 7) and absolute values (|2x - 3| < 5, |x + 1| >= 2). The answer is given in interval notation and on a number line. Equations with no solution, or every number as a solution, are spotted.

Quadratics — three methods: factoring by the ac method (split the middle term, then group), completing the square, and the quadratic formula with the discriminant explained, exact surds and complex roots. Beside the steps: the standard, vertex and factored forms, the vertex, axis and intercepts, and a plot.

Four views — sliders for a, b and c (or type exact values) and four panels that move together: the standard, vertex and factored forms (factored form greys out when there are no real roots), the parabola with its vertex, axis, roots and y-intercept, a sentence in words ("opens downward, maximum 13.48 at t = 1.53, two real roots"), and a table of values with first and second differences. Drag the purple vertex: a stays, and b = −2ah and c = ah² + k follow. The second differences are always 2a × (step)².

Roots & Δ — fix a and b and slide c. The parabola rises, its two roots slide together, meet when Δ = b² − 4ac = 0 (at c = b²/4a) and leave the real line; a second panel plots the roots in the complex plane, with faint dots for every c on the slider, so you see them travel along the real axis and then split vertically into a conjugate pair p ± qi.

Sign charts — a polynomial or rational inequality such as x^2 - x - 6 > 0 or (x - 1)/(x + 2) >= 0: everything is moved to one side, the critical points (zeros of the numerator, and of the denominator, where the expression is undefined) split the line into intervals, a test value in each fills a table with the sign of every factor and of the whole, and the solution is given in interval notation and on a number line. Zeros are included for ≤ and ≥; points where the denominator is zero never are.

Polynomials — long division and synthetic division revealed a column at a time, the remainder and factor theorems, the rational-root candidates ±p/q tested one by one, and factoring that names each pattern it uses (common factor, difference of squares, sum and difference of cubes, perfect squares, quadratic in x², grouping, a rational root).

Transformations — pick a parent function (x², |x|, √x, x³, 1/x, 2ˣ, sin x) and slide a, b, h and k in y = a·f(b(x − h)) + k; the original stays on the plot and each change is described.

Simultaneous — two linear equations by elimination or substitution, with both lines plotted and the parallel and same-line cases explained.

Practice — problems by type and level (linear, inequalities, quadratics, pairs of equations). Answers are checked by value, so x = 3/2, 1.5 and 3/2 all count, roots can be listed in any order or with ±, and inequalities can be written as x >= 3 or [3, ∞). The score and recent history stay in this browser.